Showing papers by "Jeff Erickson published in 1996"

New lower bounds for Hopcroft's problem

Abstract: We establish new lower bounds on the complexity of the following basic geometric problem, attributed to John Hopcroft: Given a set ofn points andm hyperplanes in $$\mathbb{R}^d $$ , is any point contained in any hyperplane? We define a general class ofpartitioning algorithms, and show that in the worst case, for allm andn, any such algorithm requires time Ω(n logm + n2/3m2/3 + m logn) in two dimensions, or Ω(n logm + n5/6m1/2 + n1/2m5/6 + m logn) in three or more dimensions. We obtain slightly higher bounds for the counting version of Hopcroft's problem in four or more dimensions. Our planar lower bound is within a factor of 2O(log*(n+m)) of the best known upper bound, due to Matousek. Previously, the best known lower bound, in any dimension, was Ω(n logm + m logn). We develop our lower bounds in two stages. First we define a combinatorial representation of the relative order type of a set of points and hyperplanes, called amonochromatic cover, and derive lower bounds on its size in the worst case. We then show that the running time of any partitioning algorithm is bounded below by the size of some monochromatic cover. As a related result, using a straightforward adversary argument, we derive aquadratic lower bound on the complexity of Hopcroft's problem in a surprisingly powerful decision tree model of computation.

Lower bounds for fundamental geometric problems

Abstract: We develop lower bounds on the number of primitive operations required to solve several fundamental problems in computational geometry. For example, given a set of points in the plane, are any three colinear? Given a set of points and lines, does any point lie on a line? These and similar questions arise as subproblems or special cases of a large number of more complicated geometric problems, including point location, range searching, motion planning, collision detection, ray shooting, and hidden surface removal. Previously these problems were studied only in general models of computation, but known techniques for these models are too weak to prove useful results. Our approach is to consider, for each problem, a more specialized model of computation that is still rich enough to describe all known algorithms for that problem. Thus, our results formally demonstrate inherent limitations of current algorithmic techniques. Our lower bounds dramatically improve previously known results and in most cases match known upper bounds, at least up to polylogarithmic factors. In the first part of the thesis, we develop lower bounds for several degeneracy-detection problems, using adversary arguments. For example, we show that detecting colinear triples of points requires $\Omega(\eta\sp2$) sidedness queries in the worst case. Our lower bound follows from the construction of a set of points in general position with several "collapsible" triangles, any one of which can be made degenerate without changing the orientation of any other triangle. Using similar techniques, we prove lower bounds for deciding, given a set of points in $\IR\sp{\rm d}$, whether any d + 1 points lie on a hyperplane, whether any d + 2 points lie on a sphere, or whether the convex hull of the point is simplicial. In the second part, we consider offline range searching problems, which are usually solved using geometric divide-and-conquer techniques. To study these problems, we introduce the class of partitioning algorithms. We prove that any partitioning algorithm requires $\Omega(\eta\sp{4/3}$) time to detect point-line incidences in the worst case. Using similar techniques, we prove an $\Omega(\eta\sp{4/3}$) lower bound for deciding if a set of points lies entirely above a set of hyperplanes in dimensions five and higher.

New Toads and Frogs Results

Abstract: We present a number of new results for the combinatorial game Toads and Frogs. We begin by presenting a set of simplification rules, which allow us to split positions into independent components or replace them with easily computable numerical values. Using these simplication rules, we prove that there are Toads and Frogs positions with arbitrary numerical values and arbitrarily high temperatures, and that any position in which all the pieces are contiguous has an integer value that can be computed quickly. We also give a closed form for the value of any starting position with one frog, and derive some partial results for two-frog positions. Finally, using a computer implementation of the rules, we derive new values for a large number of starting positions.

New lower bounds for convex hull problems in odd dimensions

Abstract: We show that in the worst case, Ω(ndd/2e−1 +n logn) sidedness queries are required to determine whether the convex hull of n points in Rd is simplicial or to determine the number of convex hull facets. This lower bound matches known upper bounds in any odd dimension. Our result follows from a straightforward adversary argument. A key step in the proof is the construction of a quasi-simplicial n-vertex polytope with Ω(ndd/2e−1) degenerate facets. While it has been known for several years that d-dimensional convex hulls can have Ω(nbd/2c) facets, the previously best lower bound for these problems is only Ω(n logn). Using similar techniques, we also obtain simple and correct proofs of Erickson and Seidel’s lower bounds for detecting affine degeneracies in arbitrary dimensions and circular degeneracies in the plane. As a related result, we show that detecting simplicial convex hulls in Rd is dd/2esum-hard in the sense of Gajentaan and Overmars.

New lower bounds for halfspace emptiness

Abstract: The author derives a lower bound of /spl Omega/(n/sup 4/3/) for the halfspace emptiness problem: given a set of n points and n hyperplanes in R/sup 5/, is every point above every hyperplane? This matches the best known upper bound to within polylogarithmic factors, and improves the previous best lower bound of /spl Omega/(nlogn). The lower bound applies to partitioning algorithms in which every query region is a polyhedron with a constant number of facets.